# Symmetry properties of generalized graph truncations

## TL;DR

This paper studies how symmetry properties are preserved or altered when constructing new graphs through generalized truncation, replacing vertices with smaller graphs, and applies findings to classify certain symmetric cubic graphs.

## Contribution

It provides a detailed analysis of symmetry transfer in generalized truncations and uses this to classify cubic vertex-transitive graphs of small girth.

## Key findings

- Symmetry properties depend on both original graphs involved.
- Classification of cubic vertex-transitive graphs with girths 3, 4, and 5.
- Results facilitate understanding of symmetry in complex graph constructions.

## Abstract

In the generalized truncation construction, one replaces each vertex of a $k$-regular graph $\Gamma$ with a copy of a graph $\Upsilon$ of order $k$. We investigate the symmetry properties of the graphs constructed in this way, especially in connection to the symmetry properties of the graphs $\Gamma$ and $\Upsilon$ used in the construction. We demonstrate the usefulness of our results by using them to obtain a classification of cubic vertex-transitive graphs of girths $3$, $4$, and $5$.